Super Riemann surfaces and fatgraphs

Our goal is to describe superconformal structures on super Riemann surfaces (SRS), based on data assigned to a fatgraph. We start from the complex structures on punctured $(1|1)$-supermanifolds, characterizing the corresponding moduli and the deformations using Strebel differentials and certain \v{C}ech cocycles for a specific covering, which we reproduce from a fatgraph data, consisting of $U(1)$-graph connection and odd parameters at the vertices. Then we consider dual $(1|1)$-supermanifolds and related superconformal structures for $N=2$ super Riemann surfaces. The superconformal structures $N=1$ SRS are computed as the fixed points of involution on supermoduli space of $N=2$ SRS.

1. Introduction 1.1.Some history and earlier results.The geometry of moduli spaces of (punctured) Riemann surfaces has been a central topic in modern mathematics for many years.Since the 1980s, string theory served as a significant source of ideas in studying moduli spaces.For a proper description of string theory, one has to consider certain generalizations of moduli spaces related to the fact that strings, while propagating, should carry extra anticommutative parameters, thus generating what is known as superconformal manifold as introduced by M.A. Baranov and A.S. Schwarz [1] or super Riemann surface (SRS) as independently introduced by D. Friedan [2] (see also [3], [4], [5], [6], [7], and [8] for a review).It turned out that such spaces' geometry is quite involved, e.g., [9].An important task is, of course, related to the parametrization of such supermoduli.
There are several ways of looking at the parametrization problem.For example, one could deal with supermoduli spaces of punctured Riemann surfaces with the negative Euler characteristic from the point of view of higher Teichmüller theory as a subset in the character variety for the corresponding supergroup.In the case of original moduli spaces using methods of hyperbolic geometry R. Penner described coordinates in the universal cover of moduli space, the Teichmüller space, as the subspace of the character variety of P SL(2, R), so that the corresponding Riemann surfaces appear here from the uniformization point of view as a factor of the upper half-plane by the element of the related character variety, i.e., the Fuchsian subgroup [10].
The action of the mapping class group in these coordinates is rational.It could be described combinatorially using decorated triangulations or dual objects, known as metric fatgraphs or ribbon graphs for the corresponding Riemann surfaces.Thus constructed coordinates were generalized to the case of reductive groups [11].The Date: September 26, 2023.
supergroup case yet remained a mystery until recently.In [12], [13], [14], such coordinates were constructed in the framework of the higher Teichmüller spaces associated to supergroups OSp(1|2) and OSp(2|2), which correspond to the Teichmüller spaces N = 1 and N = 2 SRS.The desired N = 1 and N = 2 SRS could be reconstructed using the elements of character variety via the appropriately modified uniformization approach [6], [15].
There is a different, more "hands-on" approach to the moduli spaces of punctured Riemann surfaces, where one can see directly the transition functions for the corresponding complex structures, which we discuss in more detail below.One can start from the parameterization of moduli spaces by the so-called Strebel differentials, which again can be described using metric fatgraphs [21].That approach allowed (see [22]) to "glue" the Riemann surface explicitly by constructing transition functions.
In this paper, we want to generalize this construction in the case of super Riemann surfaces.We start by describing the moduli space of (1|1)-supermanifolds.This result also describes the moduli space of N = 2 super Riemann surfaces.Finally, we study the moduli space of N = 1 super Riemann surfaces using the fact that this space can be obtained as a set of fixed points of the involution of the space of (1|1)-supermanifolds constructed in [7].
We would also like to mention recent progress in studying supermoduli spaces from various perspectives.While our approach deals with real parametrization of supermoduli in parallel to work on super-Teichmüller theory [12], [13], [14], a lot of exciting features of supermoduli are related to the algebro-geometric description.The main result of Donagi and Witten [16] that supermoduli space is not projected led to a renewed interest in the subject in the modern era and supergeometry in general.One can mention recent works of Felder, Kazhdan, and Polishchuk dealing with the Schottky approach for supermoduli [17] as well as the general treatment of supermoduli spaces as Deligne-Mumford stacks [18].Some other recent results, which use both real and complex geometry points of view, are related to enumerative invariants related to supermoduli [19], [20].
1.2.The structure of the paper and main results.In Section 2 we review basic notions related to (1|1)-supermanifolds, N = 1 and N = 2 Super-Riemann surfaces (SRS).We devote special attention to the punctured N = 1 SRS with two puncture classes corresponding to various spin structure choices: Ramond (R) and Neveu-Schwarz (NS).
In Section 3, we define two instrumental objects which come from geometric topology.The first object is a fatgraph (or ribbon graph).This graph is homotopically equivalent to the punctured surface with the cyclic ordering of half-edges at every vertex, which comes from the orientation of the surface so that each puncture is associated with a particular cycle on the graph.The second object is a spin structure on the fatgraph, making it a spin fatgraph.We describe spin structures as the classes of orientations on fatgraphs based on the works [12], [13], [14], where N = 1, N = 2 SRS were studied from a uniformization perspective.This construction allows distinguishing boundary components of such spin fatgraph, separating them into two sets based on comparing their orientation and the orientation induced by the surface.Those two sets correspond to NS and R punctures in the uniformization picture.
Section 4 is devoted to an important construction allowing us to relate the data assigned to the fatgraphs to the theory of moduli of Riemann Surfaces, following [22], [21].Namely, we explicitly describe the moduli spaces of moduli spaces of surfaces F c with marked points, using special covering {U v , V p }, with one neighborhood U v for every vertex v and V p for every puncture p.The set of {U v } has only double overlaps is a punctured surface.U p overlaps with all U v for all the vertices surrounding the puncture.To construct the corresponding transition functions w ′ = f v ′ ,v (w), y = f p,v (w) on overlaps, we consider the fatgraph with one positive number per every edge, producing the metric fatgraph.Then we attach the infinite stripe to the edge, with the width being the corresponding parameter.The transition functions rise from gluing stripes corresponding to edges into neighborhoods U v , with the width being a positive parameter assigned to the edge.The key ideas of this description, which is due to Kontsevich [21] and further elaborated by Mulase and Penkava [22], lies within the theory of Strebel differentials.These are holomorphic quadratic differentials on a punctured surface with certain extra conditions.One can reconstruct the metric fatgraph and the corresponding complex structure for every Strebel differential so that their zeroes define the vertices of the fatgraphs, and the order of zero determines the valence of the corresponding vertex.At the same time, the punctures correspond to their double poles.All this can be summarized in the fact that Strebel differentials parametrize the trivial R s + -bundle over the moduli space of Riemann surfaces with s punctures.
In Section 5, we use this fatgraph description to characterize the moduli space of (1|1)-supermanifolds with punctures: we use the term "puncture" for marked points or (0|1)-divisors assigned to marked points on the underlying Riemann surface.At first, we consider the split (1|1)-supermanifolds, which can be viewed as Riemann surfaces with line bundle L over it.The corresponding moduli space can be then described by the flat U(1) connections on the corresponding metric fatgraphs with zero monodromies around the punctures, accompanied by a fixed divisor at punctures, one for every degree.
Next, we describe this construction's deformation by expressing the tangent bundle's odd parts to the corresponding moduli space as Čech cocycles on the Riemann surface F .These cocycles lead to the infinitesimal deformations of the transition functions, which could be continued beyond the infinitesimal level.
Parametrizing such Čech cocycles is a nontrivial problem, which, however, can be solved in the case when deg(L) = 1 − g − n − r/2, where n is the number of point punctures and r is the (even) number of (0|1)-divisor punctures, and g is a genus.In this case, the corresponding cocycles can be characterized by the ordered sets of complex odd parameters for every vertex, where the number of parameters in each set depends on the valence of the vertex.This is roughly twice more parameters than needed, so there are equivalences between complex structures constructed in such a way.We characterize those equivalences explicitly using sections of the appropriate line bundles.
Thus the fatgraph description of the split case, together with the parametrization of cocycles, immediately leads to complete parametrization of the complex structures of (1|1)-supermanifolds with such degree.
We note that on the level of uniformization, this is an important subclass of supermanifolds obtained in [13], corresponding to flat connections with zero monodromies around punctures.
In Section 6, we use the results of Dolgikh, Rosly, and Schwarz [7], who explicitly described the equivalence between N = 2 super Riemann surfaces and (1|1)supermanifolds, expressing the transition functions of N = 2 SRS using the transition functions for (1|1)-supermanifolds obtained in Section 5.
In Section 7, we first discuss the involution on the moduli space of N = 2 SRS, such that the fixed points of this involution are N = 1 SRS.We then describe the split case, characterizing various choices of the corresponding line bundle using spin structures on the fatgraph, thus looking at the corresponding supermoduli space with the given assignment of R and NS punctures as a 2 2g covering space over moduli space of punctured Riemann surfaces.We then apply the involution to the deformations, first on the infinitesimal level and then continuing beyond, using the superconformal condition.This eventually leads to our main Theorem 7.2 which describes deformations of N = 1 SRS.
2. (1|1)-supermanifolds, N = 1 and N = 2 Super-Riemann Surfaces and Superconformal Transformations 2.1.Super Riemann surfaces and superconformal transformations.We remind that a complex supermanifold of dimension (1|1) (see, e.g., [23]) over some Grassmann algebra S is a pair (X, O X ), where X is a topological space and O X is a sheaf of supercommutative S-algebras over X such that (X, O red X ) can be identified with a Riemann surface (where O red X is obtained from O X by quoting out nilpotents) and for some open sets U α ⊂ X and some linearly independent elements {θ α } we have We will also refer to (X, O red X ) as a base manifold.These open sets U α serve as coordinate neighborhoods for supermanifolds with coordinates (z α , θ α ).The coordinate transformations on the overlaps U α ∪ U β are given by the following formulas z α = f αβ (z β , θ β ), θ α = ψ αβ (z β , θ β ), where f αβ , ψ αβ are even and odd functions correspondingly.A super Riemann surface (SRS) Σ [6], [8] over some Grassmann algebra S is a complex supermanifold of dimension 1|1 over S, with one more extra structure: there is an odd subbundle D of T Σ of dimension 0|1, such that for any nonzero section D of D on an open subset U of Σ, D 2 is nowhere proportional to D, i.e. we have the exact sequence: One can pick the holomorphic local coordinates in such a way that this odd vector field will have the form f (z, θ)D θ , where f (z, θ) is a non vanishing function and: Such coordinates are called superconf ormal.The transformation between two superconformal coordinate systems (z, θ), (z ′ , θ ′ ) is determined by the condition that D should be preserved, namely: Locally one obtains: so that the constraint on the transformation emerging from the local change of coordinates is 2.2.N = 2 super Riemann surfaces.N = 2 super Riemann surfaces (N = 2 SRS) is a generalization of super Riemann surfaces, being a supermanifold of dimension (1|2) with extra structure.Its tangent bundle has two subbundles D + and D − , so that each of them are integrable, meaning that if D ± are nonvanishing sections of D ± , we have for some functions a and b.At the same time, the direct sum D + ⊕ D − is nonintegrable, so that [D + , D − ] is a basis for the tangent bundle.Namely, for N = 2 super Riemann surface Σ one has the following exact sequence: As in the case of super Riemann surfaces one can show that there exist superconformal coordinates in which locally D + and D − are generated by: It turns out, there is an equivalence between (1|1) supermanifolds and N = 2 SRS as it was estabished by Dolgikh, Rosly and Schwarz [7].One can notice that there is an involution θ + ↔ θ − .The corresponding complex (1|1) supermanifold constructed from the N = 2 SRS after the involution is of course generally a different one and it is called dual.In fact, such a dual supermanifold turns out to be a supermanifold of (0|1) divisors of the original one.The self-dual (1|1) supermanifolds are of course N = 1 super Riemann surfaces.
We will discuss these questions in more detail later in the text.

2.3.
Punctures: Ramond and Neveu-Schwarz.Let us now discuss the types of punctures on N = 1 super Riemann surface.
The NS puncture is a natural generalization of the puncture of ordinary Riemann surface, and can be considered as any point (z 0 , θ 0 ) on the super Riemann surface.Locally one can associate to it a (0|1)-dimensional divisor of the form z = z 0 − θ 0 θ, which is the orbit with respect to the action of the group generated by D, and this divisor uniquely determine the point (z 0 , θ 0 ) due to the superconformal structure.
Let us consider the case when the puncture is at (0, 0) locally.In its neighborhood let us pick a coordinate transformation z = e w , θ = e w/2 η, (2.8) such that the neighborhood (without the puncture) is mapped to a supertube with w sitting on a cylinder w ∼ w + 2πi, and D θ becomes (2.9) Hence (w, η) are superconformal coordinates, and we have the full equivalence relation given by The case of Ramond puncture is a whole different story.On the level of super Riemann surfaces, the associated divisor is determined as follows.In this case, we are looking at the case when the condition that D 2 is linearly independent of D is violated along some (0|1) divisor.Namely, in some local coordinates (z, θ) near the Ramond puncture with coordinates (0, 0), D has a section of the form We see that its square vanishes along the Ramond divisor z = 0.One can map the neighborhood patch to the supertube using a different coordinate transformation z = e w , θ = η, (2.11) those coordinates on the supertube will be superconformal, since Notice that the identifications we have to impose on (w, η) now become: To describe Ramond punctures globally, consider a subbundle D is generated by such operators D * η , for the Ramond punctures p 1 , p 2 , . . .p n R .We have exact sequence:

14)
where P = n R i=1 P i is a divisor where D 2 = 0 mod D. In the split case T Σ| X = T X ⊕ G, dividing the tangent space to T Σ into even and odd parts, which one can identify with D 2 ⊗ O(P)and D correspondingly.Also, notice that after reducing it to the base manifold leading to the fact that there should be even number of such punctures, known as Ramond or simply R punctures.We refer to the Section 4.2.2. of [8] for more details.

Fatgraphs and spin structures
From now on we will consider Riemann surfaces of genus g with s punctures (s > 0) and negative Euler characteristic, which we will denote as F s g or simply F .The corresponding closed version will be denoted as F c .
Consider the fatgraph τ , corresponding to an s-punctured surface F .This is a graph, which is homotopically equivalent to F , with cyclic orderings on half-edges for every vertex [10] induced by the orientation of the surface.
Let τ 0 , τ 1 denote the set of vertices and edges of τ respectively.Let ω be an orientation on the edges τ 1 ⊂ τ .As in [12], we define a fatgraph reflection at a vertex v of (τ, ω) to reverse the orientations of ω on every edge of τ incident to v. Definition 3.1.We define O(τ ) to be the equivalence classes of orientations on a trivalent fatgraph τ spine of F , where the equivalence relation is given by ω 1 ∼ ω 2 iff ω 1 and ω 2 differ by finite number of fatgraph reflections.It is an affine H 1 -space where the cohomology group H 1 := H 1 (F ; Z 2 ) acts on O(τ ) by changing the orientation of the edges along cycles.
In [12,13,24] various realizations of the spin structures on the surface F , characterized by a trivalent fatgraph τ are described.Those results can be easily generalized to a fatgraph with vertices of any valence.
In fact, following [25], a spin structure can be characterized by a quadratic form q : H 1 (F ; Z 2 ) −→ Z 2 so that for any cycles a, b one has q(a + b) = q(a) + q(b) + a • b where a • b denotes the intersection form.
The space of all orientation classes is an affine H 1 -space.Really, fix a fatgraph τ , we denote by o ω (e) the orientation of the edge e ∈ τ in the orientation ω.We define which defines an element in H 1 (F ; Z 2 ).
Proposition 3.2.[12,13] The set of spin structures is isomorphic to the space of quadratic forms Q(F ) on H 1 (F ; Z 2 ), and also isomorphic to O(τ ), as affine H 1 -spaces.
This leads to the following important consequence.
Theorem 3.3.[13] Given an oriented simple cycle φ ∈ π 1 (F ) homotopic to a path on the fatgraph τ with orientation class [ω], the corresponding quadratic form is given by where L φ (resp.R φ ) is the number of left (resp.right) turns of γ on the fatgraph τ , and N φ (resp.N γ ) is the number of edges of τ such that γ and ω have the same (resp.opposite) orientation.
If we talk about the paths corresponding to the boundary cycles on the fatgraph, q([γ]) = (−1) k , where k is a number of edges with orientation opposite to the canonical orientation of γ.One can identify them with R and NS punctures in uniformization picture [14], so that k is even for R punctures and odd for NS punctures.
In fact there is another way of thinking about the spin structures, using graph connections [26], [10], [27].Definition 3.4.[10] Let G be a group.A G-graph connection on τ is the assignment g e ∈ G to each oriented edge e of τ so that g e = g −1 e if e is the opposite orientation to e. Two assignments {g e }, {g ′ e } are equivalent iff there are t v ∈ G for each vertex v of τ such that g ′ e = t v g e t −1 w for each oriented edge e ∈ τ 1 with initial point v and terminal point w.
Therefore, we obtain the following description of spin structures.Corollary 3.5.[13] The space of spin structures on F is identified with Z 2 -graph connections on a given fatgraph τ of F .
We will refer to the fatgraph with the associated spin structure/Z 2 -graph connection as spin fatgraph.

Complex structures and Strebel differentials
4.1.Gluing of Riemann surfaces.Consider the fatgraph τ corresponding to an spunctured surface F of genus g.Let us assign a positive real parameter L j associated to every edge j.We will refer to the resulting object as a metric fatgraph.
It is known from the works of Penner (the so-called convex hull construction) that the fatgraphs with valence of every vertex greater ore equal to 3 or dual ideal cell decompositions of Riemann surfaces describe the mapping class group-invariant cell decomposition of the decorated Teichmüller space (see e.g.[10], [28]), a universal cover of R s + ⊗ M g,s , where M g,s is a moduli space of Riemann surfaces of genus g with s marked points.Then the trivalent fatgraphs correspond to the higher-dimensional cells of dimension 6g − 6 + 3s.
An important problem is how to reproduce inequivalent complex structures based on the data of metric fatgraphs.In an important work of Mulase and Penkava [22], based on earlier ideas of Kontsevich [21], allows to construct the appropriate covering of a Riemann surface and the transition functions, associated with a given fatgraph, thus exhausting all possible complex structures.Let us have a look at those in detail.
Fixing an orientation on τ , we consider a neighborhood U v with coordinate w corresponding to the fixed m-valent vertex v, so that the vertex is placed at the point w = 0.One can describe that neighborhood by considering stripes glued together via formula if all m edges are pointing out from vertex v.In the case if one or more of them is pointing towards vertex v, we substitute the above formulas by One can construct such coordinate patches around every such vertex.The overlaps U v ∩ U v ′ are described by the corresponding stripes associated to the edge j of the fatgraph runnig between v and v ′ .Note, that there is no triple intersections on such a punctured surface and that the vertices of the fatgraph belong to the boundary of the intersections.
Let us look at the transition functions on the overlaps between two such coordinate neighborhoods U v , U v ′ around neighboring vertices v and v ′ , assuming the edge is pointing from v to v ′ .We note that both coordinates w and w ′ are expressed in terms of z j in the following way: where c j , c ′ j are mth and m ′ th roots of unity.The resulting overlap coordinate transformation f v ′ v between patches is given by the following formula: where −iπ/2 < arg(w m/2 )< iπ/2.That completely describes the transition functions between charts for the punctured Riemann surface F .
Note that if the consecutive edges L 1 , L 2 , . . ., L n correspond to boundary piece of the fatgraph associated with puncture p, one can glue the following coordinate neighborhood V p with coordinate y covering the puncture: so that one glues together top or bottom part of the stripes based on orientation and x-variable is on a cylinder x ∼ x + 2πi.Suppose the k-th strip is glued to the vertex v k with coordinate w k as above, then the transition function f pv is given by: ) .(4.7) In the following we will adopt the following notation for L-parameters: if the edge connects two vertices v, v ′ we will denote the corresponding parameter as L v,v ′ .4.2.Strebel differentials.An important object in the constructions of [21], [22] are the Strebel differentials, the quadratic meromorphic differentials with special properties.A nonzero quadratic differential is a holomorphic section µ of K ⊗2 , where K stands for canonical bundle on F .It defines a flat metric on the complement of the discrete set of its zeroes, written in local coordinates as |µ(z)|dzdz, where µ = µ(z)dz 2 .
A horizontal trajectory of a quadratic differential is a curve along which µ(z)dz 2 is real and positive.The Strebel differential is the one for which the union of nonclosed trajectories has measure zero.Non-closed trajectories of a given Strebel differential decompose the surface into the maximal ring domains swept out by closed trajectories.These ring domains can be annuli or punctured disks.All trajectories from any fixed maximal ring domain have the same length, the circumference of domain.
The following Theorem is due to Strebel: For every such Strebel differential one can construct the covering associated to the corresponding fatgraph, described in the previous section and vice versa, so that in the charts U v , Strebel differential µ has the following explicit form: It also has pole of order 2 at punctures, so that in y-coordinates for each neighborhood V p the differential looks as follows: One can formulate then the following Theorem.In this paper we do not need more properties of Strebel differentials, however, we refer reader to [22], as well as original source [29] for more information.
5. Complex structures on (1|1) supermanifolds 5.1.Split case.Let us consider the punctured Riemann surface glued as in previous subsection using metric fatgraph and overlapping neighborhoods U v corresponding to vertices.To construct the cooordinate transformations for a split (1|1)-supermanifold SF with such a base complex manifold, one has to consider a line bundle L over F c .Then the coordinate transformations for the coordinates (w ′ , ξ ′ ), (w, ξ), (y, η) corresponding to neighborhoods U v , U v ′ , V p of vertices v, v ′ and puncture p are given by the following formulas: where g v ′ ,v , g p,w is the holomorphic function, serving as a transition function of bundle L. The collection {g v ′ ,v , g v,p } generates a Čech cocycles representing the Picard group of F c , i.e.Ȟ1 (F, O * ), if the following constraint on g v ′ ,v and {g v,p } is imposed around the given puncture p: Then the following Proposition holds.Proposition 5.1.When L is degree 0 over F c , the fatgraph data describing it is a U(1) graph connection with a trivial monodromy around every boundary piece.
Proof.Notice, that one can choose g v ′ v to be constant functions with values on a unit circle, which on the level of fatgraph is described by U(1)-graph connection, so that g v,v ′ = e ih v,v ′ , where h v,v ′ ∈ R .Indeed the corrresponding holomorphic equivalences for the corresponding Čech cocyle reduce to constant U(1) gauge transformations at the vertices.However, according to the condition (5.3) that we imposed, we have to have g v 1 ,v 2 g v 2 ,v 3 . . .g v n−1 ,vn g vn,v 1 = 1, which is exactly the trivial monodromy condition.
In order to describe any line bundle of degree d, one has to do the following.First, choose a fixed divisor of degree d, say a linear combination of puncture points.Then, multiplying it on appropriate bundle of degree 0, one can reproduce the original bundle.Since we described the moduli spaces of degree 0 bundles in a Proposition 5.1 above, we can now characterize split punctured supermanifolds.
• Fixed divisor M of degree d, which is a linear combination of puncture points.
The data above determines the complex split (1|1)-supermanifold corresponding to the line bundle of degree d on F .For a fixed divisor M, metric fatgraphs with U(1) connections describe the moduli space of split (1|1)-supermanifolds.
5.2.Infinitesimal Deformations and various types of punctures.As usual, the infinitesimal deformations of the above formulas leading to generic non-split structure are described by H 1 (SF c , ST ), where ST is a tangent bundle of SF c , where SF c is a split (1|1)-supermanifold which we discussed in the previous section.Since we are deforming the split case one can describe infinitesimal deformations ρ ∈ H 1 (SF c , ST ) using Čech cocycles, i.e. in coordinates (ξ, w) of where the indices v, v ′ and p, v here mean that the corresponding elements are the corresponding Čech cocycles considered on the intersections U v ∩ U v ′ , and V p ∩ U v .Now we need to specify the behavior at the punctures to describe the cocycles ρ leading to deformations of SF in terms of cocyles on F c .
There are two types of punctures we want to consider: • puncture as a (0|1)-dimensional divisor on SF c .We denote the number of such punctures as r.
• puncture as a (0|0)-dimensional divisor, or in other words just a point on SF c .We denote the number of such punctures as n.
Let T be the tangent bundle of F c , D n+r be the divisor corresponding to sum of all points on F c corresponding to punctures, and D n is the sum of the ones corresponding to point-punctures on SF .Let us look in detail at the components of (5.4): where Ž1 is the notation for Čech cocycles of degree 1. Note, that we need to impose the constraints on cocycles on where s = v, u, α, β.Here u-and v-terms correspond to the deformations of the original manifold F , and notice, that we already incorporated the moduli for the base manifold F and the line bundle L in the formulas (5.1).The odd deformations, provided by the cycles α, β give the following deformations for the upper line of (5.1) which describes (in the first order in complex parameters)all possible complex structures on the punctured supermanifold.
If we remove the infinitesimality condition, formulas above will be deformed.Let us formulate it in a precise form.(1) Consider the following data: • A metric fatgraph with a U(1)-connection with trivial monodromy around boundary pieces, a fixed divisor which is a iinear combination of puncture points of degree d, which defines a split punctured (1|1) supermanifold determined by base Riemann surface F and line bundle L.
• Čech cocycles where D n+r is the divisor corresponding to sum of all s = n + r punctures on the closed surface F c , D n is the sum of the certain subset of the set of punctures, and the cohomology classes of {b k } {a k } form a basis in the corresponding cohomology spaces.This data gives rise to a family of complex structures on SF , the (1|1)supermanifold with n point punctures and r (0|1)-divisor punctures, so that the transition functions on SF are given by the following formulas on the overlaps {U v ∩ U v ′ }: where g , are holomorphic functions on the overlaps, depending on the parameters σ α k and σ β k such that: where {f v ′ v }, {g v ′ ,v } define the split supermanifold with the line bundle L and s punctures so that in the first order in {σ α k } and {σ (2) Let us fix the choice of transition functions in (5.8), for every metric fatgraph τ with the U(1)-connection, divisor of degree d, and the odd data given by the cocycles β, α on F c .The complex structures constructed in such a way are inequivalent to each other, and the set of such complex structures constructed by varying τ and the data on it, form a dense subset of maximal dimension in the moduli space of punctured (1|1) supermanifolds with underlying line bundles of degree d.
Proof.Let us look at the formulas (5.8) as generic ones, for arbitrary holomorphic functions {α v ′ v }, {β v ′ v } on overlaps.There is a finite number of odd parameters which parametrize all {α v ′ ,v }, {β v ′ ,v } corresponding to inequivalent complex structures.Expanding the formulas (5.8) in terms of these parameters we obtain that in the linear order as in the infintesimal case.Conversely, since α, β represent the tangent space to the moduli space of complex structures, parameters σ α , σ β serve as coordinates there.Considering the corresponding 1-parametric subgroups generated by α, β we obtain formulas from (5.8).The fact that the cohomologically equivalent cocycles lead to the equivalent complex structures is justified by dimensional reasons.
It is, however, nontrivial to explicitly parametrize those cocycles α, β.In the next subsection we will analyze the special case of supermanifolds with the line bundle L of negative degree.

(1|1)-supermanifolds with deg
It is not easy to explicitly parametrize cocycles α, β from a fatgraph data if one does not fix a degree.From now on, we will be interested in the case when deg(L) = 1 − g − k, where s ≥ k ≥ 0 on F c .Assuming that the number of divisor punctures is even and setting k = r/2, both bundles Let us be generic enough first and characterize the cycles in Π Ž1 (F c , L ⊗ O(−D n )) where s ≥ k = g − 1 − deg L ≥ 0, using the data from the fatgraph.To do that, we define a cocycle ρ, a representative of Π Ȟ1 (F c , L ⊗ O(−D n )) as follows: are the polynomials with odd coefficients, assigned to each fatgraph vertex v of degree at most Then the following proposition holds.
(1) The cycles (5.9) are uniquely defined by the numbers σ v at the fatgraph vertices, thus forming a complex vector space of dimension 4g−4+2s.
Proof.To prove part (1) one just has to count the number of vertices and parametres at verrtices.An elememtary Euler characteristic computation shows that 2g − 2 + s = j≥3 j 2 − 1 V j (τ ), (5.11)where V j (τ ) is the number of j-valent vertices in τ .Notice, that for a j-valent vertex v, we have exactly j −2 odd parameters from the expansion of σ v (w), which immediately leads to the necessary parameter count, giving 4g − 4 + 2s.
To prove (2), on each coordinate neighborhood U v , Strebel differential µ has the form µ| Uv = w mv−2 dw 2 , and µ| Vp = − a 2 p 4π 2 dy 2 y 2 , which means that one can rewrite the formula for the cocycle (5.12)where Suppose that such cocycle is exact, namely: where γ (m−3) (w) is the Taylor expansion of γ(w) up to order m − 3. Also the identity γ p = a p µ + γ| Vp , i.e. a p µ + γ| Vp = 0 is possible only if γ| Vp has poles not greater than 2 at y = 0, or, more precisely, γ ∈ H 0 (V p , L ⊗ K 2 ⊗ O(2D n+r )).Therefore, cycles ρ and ρ are cohomologous to each other iff the relation between the parameters on the fatgraph {σ v } and {σ v } correspodingly parametrizing them is as follows: where γ ∈ H 0 (F c , L⊗K 2 ⊗O(2D n+r )) on F c , such that γ| Uv = γ(w) with poles at the punctures of F of order less or equal to 2 so that γ (m−3) (w) is the Taylor expansion of β(w) up to order m − 3. Now, to prove part (3), we need to show that such classes of cocycles form a 2g − 2 + k + n-dimensional complex space as elements of Π Ȟ1 (F c , L).For a given section γ of L⊗K 2 ⊗O(2D n+r ), the collection of the coefficients in γ (m−3) , for each vertex v form a vector in our 4g − 4 + 2s-dimensional space of σ-parameters.The space, spanned by all such vectors is a complex 2g −2+2s−k −n-dimensional space.Indeed, it cannot be of smaller dimension, since we know that dim C Ȟ1 (F c , L ⊗ O(−D n )) = 2g − 2 + k + n, at the same time it cannot be of greater dimension, since we know that the dimension of space of such meromorphic global sections of L ⊗ K 2 ⊗ O(2D n+r ) is 2g − 2 + 2s − k − n by the Riemann-Roch theorem.Now we are ready to formulate a Theorem regarding parametrization of complex structures via fatgraph data.
(1) Consider the following data associated to the fatgraph τ : • Metric structure and a U(1)-connection on τ with zero monodromy around punctures and a fixed divisor of degree d = 1 − g − r/2 at the punctures.
We will call two sets of data from (1) associated to fatgraph τ equivalent if the odd data are related as in Theorem 5.4.
Constructing transition functions f v ′ ,v and g v ′ ,v from the even fatgraph data and cocycles α, β, corresponding to r (0|1)-divisor punctures and n point punctures from the odd data, one obtains a family of complex structures on (1|1)-supermanifold in the framework of Theorem 5.3.
(2) Fixing the transition functions in (5.8) and considering one such complex structure per equivalence class of data for every fatgraph τ , we obtain a set of inequivalent complex structures, which is a dense subspace of odd complex dimension 4g − 4 + 2n + r in the space of all complex structures on (1|1)supermanifolds with base line bundle of degree d = 1 − g − r/2 and s = n + r punctures, where n is the number of point punctures and r is the number of (0|1)-divisor punctures.
Proof.The first part of data allows to construct split (1|1)-supermanifold as we know from previous sections, the odd data from the second allows to construct the corre- If we choose an orientation on the fatgraph, the formulas (see Theorem 5.3): produce the transition functions on U v ∩ U v ′ for the vertex oriented from v to v ′ .
Remark.Note, that the gauge equivalence for U(1) connection, produce the following identification.If real numbers h v,v ′ parametrize U(1) connection, then the transformations: produce equivalent configuration for infinitesimal parameters σ.In the paper [13] the uniformization version of N = 2 Teichmüller space was constructed (see also [3], [15]), which corresponds exactly to (1|1)-supermanifolds, which serves as a universal cover for the one we use here in the case of deg(L) = 1 − g − r/2.The above identifications played an instrumental role in the construction.
In the next two sections we will use the obtained results to describe transition functions for the corrresponding dual supermanifold and N = 2 super Riemann surface following [7].5.4.Dual (1|1) supermanifold.Finally, we give a desription of the concept dual (1|1) supermanifold is a supermanifold of (0|1) divisors of SF .To describe the explicit coordinates and coordinate transformations transformations on such an object one can use a very simple equation (see, e.g., [8]): (5.17 We will now substitute first equation in the second and obtain: which leads to two equations: The latter equation immediately gives the transformation for ζ: which could be simplified as follows: Now, substituting that into the equation (5.18) for a ′ we obtain: a), and simpler: One can see from the transformations we obtained that the self-dual (1|1) supermanifolds are indeed N = 1 SRS.Let us combine all that in the following theorem.
Theorem 5.6.Given the coordinate transformations (5.15) for SF , the coordinate transformations for the dual manifold SF of (0|1) divisors is given by the formulas: , where (5.20) Remark.Note, that in the case of a dual manifold, L is replaced by L −1 ⊗ T .

N=2 Super Riemann Surfaces
In this section we write down the coordinate transformations for punctured N = 2 supermanifold SF N =2 , corresponding to SF , based on the equivalence between complex structures on (1|1)-supermanifolds and superconformal structures on N = 2 supermanifolds discovered in [7].
Let us write down the transition functions between the chart with coordinates (z, θ) and chart with coordinates (u, η) on (1|1) supermanifold in the following way: u = S(z) + θV (z)ϕ(z), η = ψ(z) + θV (z), (6.1)where S(z), V (z) and ϕ(z), ψ(z) are correspondingly even and odd analytic functions.Onthe other hand, the superconformal coordinate transformations for N = 2 SRS between the charts with coordinates (z, θ + , θ − ) and (z ′ , θ ′ + , θ ′ − ) are: There is a following Theorem matching these transformations.Theorem 6.1.[7] There is one-to one correspondence between N = 2 SRS from (1|1)supermanifolds.The explicit correspondence between transition functions is given by the following formulas: Let us now describe how it works for the transition functions we introduced in the previous section.In our case: Therefore, we can write for the transition functions of SF N =2 : . This can be rewritten in a simpler way: Hence we obtain the following theorem.Theorem 6.2.Formulas (6.5) produce the transition functions describing the superconformal structure on N = 2 SRS with punctures, corresponding to (1|1)-supermanifolds with transition functions (5.15).Namely the transition function corresponding to oriented edge v, v ′ of the fatgraph, i.e. the overlap U v ∩ U v ′ is desribed by the functions ǫ ± (w), q ± (w) from (6.5).

7.
Involution and N = 1 Super-Riemann surfaces with NS and R punctures.

Involution: R vs NS punctures.
There is an involution I on the moduli space of super-Riemann surfaces, such that where D ′ ∓ is the corresponding operator after N = 2 superconformal transformation.Such an involution takes N = 2 super Riemann surface to the dual, which on the level of (1|1)-supermanifolds produces a manifold of (0|1)-divisors, which we discussed earlier.The self-dual supermanifolds are known to be N = 1 super-Riemann surfaces.
Note, that the two examples of the action of involution which we considered in this section are the only ones, which preserve the base manifold.7.2.Split N = 1 SRS.Let us now discuss split N = 2 SRS, which implies that we let cocycles α, β = 0.The involution acts on the level of transition functions as follows: Therefore, for fixed points of the involution we have . (7.5)This means that g w) where sign(v ′ , v) is the notation for the sign of the square root, so that on a resulting N = 1 SRS we have: Choice of signs for such square roots is the same as the choice of spin structure on the punctured surface.However, we already discussed that problem on the level of fatgraphs (Section 3), which leads to the following Theorem.Theorem 7.1.Consider a metric fatgraph τ with a spin structure ω provided by the orientation as discussed in Section 3.This data defines the superconformal structure on the split N = 1 SRS.For every boundary cycle on the fatgraph, corresponding to puncture p, let m p be the number of oriented edges, which are opposite to the orientation induced by the one on the surface.The corresponding puncture is Ramond or Neveu-Schwarz, depending on whether m p is even or odd.
Proof.So, let us consider the metric graph τ with orientations on edges.Our problem is to use orientations to define To do that, for each overlap we will look at the z coordinates on stripes, discussed in section 4. For given vertices v and v ′ , the transformation between z and z ′ coordinates is given by We will define the value of the ∂ z fv ′ ,v (z) = ±i in the following way.If the orientation is from vertex v to v ′ we choose the positive sign (sign(v, v ′ ) = 1) and negative otherwise (sign(v, v ′ ) = −1).One can prove that such choice does not depend on the choice of orientation for a given spin structure, namely a different choice, corresponding to a fatgraph reflection, will just result in a reflection of an odd coordinate for a given vertex.
Regarding R and NS punctures, one can deduce immediately that the statement is correct just by a simple condition that there is a natural combinatorial constraint on the punctures with m p being odd on a fatgraph (see Section 4), matching the one for Ramond punctures on a surface.Nevertheless, let us prove that directly.
For a given choice of spin structure, let us superconformally continue g , where we remind that where z is the coordinate on the consequtive stripe v k , v k+1 and v 1 , . . ., v m are consequtive vertices around the puncture.Now we obtain R and NS punctures by gluing the supertube with a proper twist of the odd variable.That will of course depend on the number m p of the edges {v i , v i+1 }, which have opposite orientation with respect to orientation induced on the cycle by the one on the surface.Note, that in terms of z-variables ∂ z fp,v (z) is a constant, so one can again make a choice of signs explicitly.
We have the following identity where positive sign is for even N and negative otherwise.In the case of m p even, we can choose {sign(p, v)} so that for sign(v, v ′ ), so that v, v ′ are neighboring vertices, so that sign(v, v ′ ) = sign(p, v)sign(p, v ′ ), thus gluing the stripes into supertube.However, this is not possible in the case of odd m p .In this case we have to assume that sign(v n , v 1 ) = −sign(p, v 1 )sign(p, v n ), thus gluing the stripes into twisted supertube corresponding to NS puncture.
Remark.One can of course superconformally transform the twisted supertube in NS puncture case into the disk, the same way we did in the introduction, thus making the corresponding cocycle {g v,p (y)} = {± ∂ y f v,p (y)} .In Ramond case this is of course impossible.We see that if p is an R puncture, . Therefore, for the bundle L we have a condition: which is possible only when n R is divisible by 2. 7.3.N = 1 SRS: non-split case.In order to construct nonsplit N = 1 SRS, we first will do it on infinitesimal level near the split N = 1 SRS.So, let us look at the formulas (6.5) when α v ′ ,v β v ′ ,v are infinitesimal: . (7.9)An invariance under simple involution D ± −→ D ∓ allows to identify α v ′ ,v and β v ′ ,v and as before g 2 v ′ ,v (w) = ∂ w f v ′ ,v (w), thus infinitesimally the transformations for the resulting N = 1 SRS on the overlap U v ∩ U v ′ is given by: w)(ξ + ρ v ′ ,v (w)), (7.10) so that the signs of are prescribed as in the Theorem 7.1, where ρ ∈ Π Ž1 (F c , L ⊗ O(−D N S )) and L 2 = T ⊗ O(−D R ), so that D R and D N S are the divisors corresponding to the sum of all NS and R punctures correspondingly.We described such cocycles using odd number decorations at the vertices of the the fatgraph in the Theorem 5.4.The formulas (7.10) are not hard to continue to full superconformal transformations for transition functions (one can obtain them by applying involution D ± −→ D ∓ invariance to the formulas (6.5) as well): v ′ ,v (w))(ξ + λ v ′ ,v (w)).Combining the parametrization data for cocyles ρ from Theorem 5.9 with the results of this section, we obtain the following omnibus Theorem, describing the dense set of superconformal structures oinside moduli space of N = 1 SRS.Theorem 7.2.Consider the following data on a fatgraph τ : (1) Metric structure.
(2) Spin structure, as equivalence class of orientations on the fatgraph.The cycles on the fatgraph, encircling the punctures are divided into two subsets, NS and R, depending on whether there is odd or even number of edges oriented opposite to the surface-induced orientation of the appropriate boundary piece of a fatgraph correspondingly.We denote the number of the corresponding boundary pices as n R and n N S .
(3) Ordered set {σ k v } k=0,...,mv−3 of odd complex parameters for each vertex v, where m v is the valence of the vertex v.
Then the following is true: (a) Data from (1) and (2) determine uniquely the split Riemann surface with n R Ramond and n N S Neveu-Schwarz punctures with the transition functions given by (7.11) one for each overlap U v ∩ U v ′ .The sign of the square root is given by the spin structure on the fatgraph, making odd coordinate a section of a line bundle L on the corresponding closed Riemann surface F c , such that L 2 = T ⊗ O(−D R ), where D R is a divisor, which is a sum of points corresponding to the Ramond punctures.
where ρ v , ρ v ′ are the meromorphic sections of L ⊗ O(−D N S ) on U v , U v ′ correspondingly, so that m v is valence of the given vertex v.The cocycles defined by configurations described by {σ v } and {σ v } are equivalent to each other if and only if σ v (w) − σv (w) = γ (m−3) (w), (7.13) for every vertex v, γ ∈ ΠH 0 (F c , L ⊗ K 2 ⊗ O(D N S + 2D R )), γ| Uv = γ(w) so that γ (m−3) (w) is the Taylor expansion of γ(w) up to order m − 3.
We call two sets of data associated to the fatgraph τ equivalent, if they are related as in (7.13).
(c) There exist a superconformal structure for N = 1 super Riemann surface SF with n R Ramond punctures and n N S Neveu-Schwarz punctures so that the superconformal transition functions on for each overlap U v ∩ U v ′ are: ) v ′ ,v (w)), (7.14)where the deformed functions f (d) To describe the non-split SRS, we fix the choice of transition functions in (c) for every metric spin fatgraph τ with the odd data from (3).We consider the set of superconformal structures constructed by picking one superconformal structure per equivalence class of data for every fatgraph τ .The points in this set represent inequivalent superconformal structures, and together they form a dense subspace of odd complex dimension 2g − 2 + n N S + n R /2 in the space of all superconformal structures with n N S Neveu-Schwarz and n R Ramond punctures associated to F .

Theorem 4 . 1 . [ 29 ]
For any connected closed Riemann surface F c with s distinct points p 1 , . . ., p s , s > 0 and genus g, s > χ(F c ) = 2 − 2g and n positive real numbers a 1 , . . ., a s there exists a unique Strebel differential on F = F c \{p 1 , . . ., p s }, whose maximal ring domains are s punctured disks surrounding p i 's with circumference a i 's.The union of non-closed trajectories of Strebel differentials together with their zeroes define a graph, embedded into a Riemann surface, thus giving to it a fatgraph structure.Every vertex of a fatgraph corresponds to the zero of the Strebel differential of degree m − 2, where m ≥ 3 is the valence of the vertex.The length of each edge gives the graph a metric structure.

Theorem 4 . 2 .
[21] Let M comb g,s denote the set of equivalence classes of connected ribbon graphs with metric and with valency of each vertex greater than or equal to 3, such that the corresponding noncompact surface has genus g and s punctures numbered by 1, . . ., s.The map M g,s × R s −→ M comb g,s which associated to the surface F c and numbers a 1 , . . .a s the critical graph of the canonical Strebel differential from Theorem 4.1 is one-to-one.
) w = a + ζξ where a, ζ are the coordinates parametrizing such a (0|1) divisor.Let us derive the formulas for the transformations of a, ζ variables, for the transformation between the charts with coordinates (a, ζ) and (a ′ , ζ ′ ), so that w (b) Part (3) of the above data allows to construct Čech cocycles on a Riemann surface F, which are the representatives of Π Ȟ1 (F c , L⊗O(−D N S )), where D N S is a divisor, corresponding to the sum of the points corresponding to NS punctures: (σ) v ′ ,v , λ (σ) v ′ ,v depend on odd parameters {σ k v }, characterizing Čech cocylce {ρ v ′ ,v }, with f (0) v ′ ,v = f v ′ ,v and in the first order in {σ k v } variables λ (σ)